Notes on Higher Order Convex Functions (VIII)

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Tiberiu Popoviciu

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T. Popoviciu, Notes sur les fonctions convexes d’ordre supérieur (VIII), Bull. de la Sect. Sci, de l’Acad. Roum, 22 (1939) no. 1, pp. 34-41 (in French). [JFM 65.0214.02].

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1939 a -Popoviciu- Bull. Sect. Sci. Acad. Roum. – Notes sur les fonctions convexes d’ordre superieur (VIII)

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1939 a -Popoviciu- Bull. Sect. Sci. Acad. Roum. - Notes on convex functions of higher order

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ROMANIAN ACADEMY

BULLETIN OF THE SCIENTIFIC SECTION

NOTE ON HIGHER ORDER CONVEX FUNCTIONS (VIII)

BY

T'IBERIU POPOVICIU

Note presented by Mr. S. Stoilov, M. v. AR, in the session of July 14, 1939.
ON THE LOCAL DEFINITION OF ORDER FUNCTIONS

n

I. Let E 1 to 1; any

= min E, has ≦ b =

max E the c:

e) of E. If, t closed it is necessarily in in soli turoot

ϵ_{i} \Rightarrow

almost-closed are

\dot{E}

of

E

is the set ... ... its derivative

E^{'}

except the ends

has

,

b

which do not belong to E. If E = E, we will say that the set E is almost closed. We will say that a subset

E_{1}

of E is a section of

E_{1}

whether it is formed by a single point or with

x_{1} \in E_{1}

,

x_{2} \in E_{1}

all points of

E

belonging to the interval (

x_{1}, x_{2}

) belong to

E_{1}^{1}

). If two sections of E have at least one point in common, their union and their intersection are still sections of E . If two sections of E have no points in common, every point of one is to the left of every point of the other. In this case, they are either separated by E , so their union is not a section, or they are two consecutive sections, so their union is still a section of E .

The neighborhood

V_{x}^{k}

from one point

x

is a section of E having at least

k

points left and at least

k

points to the right of

x

. If there is only

r < k (r > 0)

points of E to the left (right) of

x, V_{x}^{k}

must contain all these points and at least

2 k - r

points to the right (left) of

x

. In addition, the neighborhoods

V_{has}^{k}

must contain with

x_{1} \in V_{has}^{k}

all points of

F_{1}

belonging to the closed interval (

a x_{1}

). The same applies to the surrounding areas.

V_{b}^{k}

. In this definition

k

is a natural number, so if

x \in E

we have

x \in V_{x}^{k}

. In the following we only consider neighborhoods.

V_{x}^{k}

Or

x \in \dot{E}

. I, when we consider several neighborhoods

V_{x}^{k}

they are all taken for the same value of

k

. It is then useless to consider sets

E

having less than

2 k + 2

points.

2. Two neighborhoods

V_{x}^{k}

corresponding to the same point

x

have at least

2 k

common points

^{1}

). Now consider a neighborhood

V_{x_{0}}^{k}

and either

x_{1} \in V_{x_{0}}^{k}

a point to the right of

x_{0}

. Let us further assume that

V_{x_{0}}^{k}

still has at least

s \geq 0

points to the right of

x_{1}

. Consider a neighborhood

V_{x_{1}}^{k}

of

x_{1}

and let's see how many points it can have in common with

V_{x_{0}}^{k}

. We immediately see that

V_{x_{0}}^{k} \cdot V_{x_{1}}^{k}

have at least min

(s, k)

points in common to the right of

x_{1}

. If there is at least

k

points of E to the left of

x_{1}, V_{x_{0}}^{k}, V_{x_{1}}^{k}

have at least

k

common points to the left of

x_{1}

. It remains to be seen what happens if there is only

r < k

points of E to the left of

x_{1}

. In this case

s > 2 k ⟶ r

And

V_{x_{0}}^{k}, V_{x_{1}}^{k}

have at least

2 k - r

common points to the right of

x_{1}

and have in common all the points of E to the left of

x_{1}

. In any case, we can say that

V_{x_{0}}^{k}, V_{x_{1}}^{k}

have at least min

(s, k) + k + I

common points. A similar property remains if

x_{1} < x_{0}

, SO

Lemma I. If

V_{x_{0}}^{k}

is a neighborhood of a point

x_{0}

of

\dot{E}

And

V_{x_{1}}^{k}

a neighborhood of a point

x_{1}

of

V_{x_{0}}^{k}

, the sets

V_{x_{0}}^{k}, V_{x_{1}}^{k}

have at least

min (s, k) + k + I

common points, assuming that

V_{z_{0}}^{k}

has at least

s (\geq 0)

points to the right of

x_{1}

if

x_{0} < x_{1}

or to the left of

x_{1}

if

x_{1} < x_{0}

.

Corollary I. If E

n

'has no point between

x_{0}, x_{1}

, two neighborhoods

V_{x_{0}}^{k}, V_{x_{1}}^{k}

have at least

2 k

common points.
3. Now consider two neighborhoods

V_{p}^{k}, V_{q}^{k}, p \leq q

, which are not separated by E . We distinguish the following four cases:

I^{0} V_{p}^{k}, V_{q}^{k}

have at least

2 k

common points,

2^{0} V_{p}^{h}, V_{q}^{h}

have

r, k \leq r < 2 k

, common points,

3^{0} V_{p}^{k}, V_{q}^{k}

have

r, I ≦ r < k

, common points,

4^{0} V_{p}^{k}, V_{q}^{k}

have nothing in common.
When

p = q

we are in the case

1^{0}

. For cases

2^{0}, 3^{0}, 4^{0}

it is therefore necessary that

p < q

. In the case

2^{0}

be

x_{1} < x_{2} < \dots < x_{1}

the common points of

V_{p}^{k}, V_{q}^{k}

. Consider any neighborhoods

V_{x_{1}}^{k}, V_{x_{2}}^{k} \dots

,

V_{x_{j}}^{k}

. In the sequel

\begin{matrix} (I) & V_{x_{1}}^{k}, V_{x_{2}}^{k}, \dots, V_{x_{k}}^{k} \end{matrix}

two consecutive terms have at least

2 k

common points, by virtue of corollary I. If

p

coincides with a point

x_{i}, V_{p}^{k}, V_{x_{i}}^{k}

have at least

2 k

common points. Otherwise we have

p < x_{1}

Or

x_{p} < p

and the sets

V_{p}^{k}, V_{x_{1}}^{k}

Or

V_{p}^{k}, V_{x_{p}}^{k}

onit at least

2 k

common points. The same is true for

V_{q}^{k}

. If

p < q \leq x_{1}

Or

x_{γ} \leq p < q V_{p}^{k}, V_{q}^{k}

have at least

2 k

common points and we are actually in the case

1^{0}

. Let's examine the case

3^{0}

.

Let's still be

x_{1} < x_{2} < \dots < x_{r}

the common points of

V_{p}^{k}, V_{q}^{k}

and let (I) be any neighborhoods. The same considerations apply as before except that we can only assert that

V_{p}^{k}, V_{x_{1}}^{k}

Or

V_{p}^{k}, V_{x_{p}}^{k}

have at least

k + r

common points, these pairs of neighborhoods are therefore in the case

1^{0}

Or

2^{0}

. The same applies to

V_{q}^{k}

. If

p < q \leq x_{1}

Or

x \leq p < q, V_{p}^{k}, V_{q}^{k}

have at least

k + r

common points and we are actually in the case

1^{0}

Or

2^{0}

. We still have the case

4^{0}

. In this case either

d

the right end of

V_{p}^{k}

And

V_{d}^{k}

any neighborhood of

d

. We immediately see that the two neighborhoods

V_{p}^{k}, V_{k d}

and the two neighborhoods

V_{d}^{k}, V_{q}^{k}

are in the case

1^{0}, 2^{0}

Or

3^{0}

:

The previous analysis shows us that we can state
Lemma II. If

V_{p}^{k}, V_{q}^{k}, p < q

are two neighborhoods that are not separated by E, or they have at least

2 k

common points, or we can find a finite number of points

x_{1}, x_{2} \dots, x_{m}

of E such that, if

V_{x_{i}}^{k}

are any neighborhoods, in the sequence

V_{p}^{k}, V_{x_{1}}^{k}, V_{x_{2}}^{k}, \dots, V_{x_{m}}^{k}, V_{q}^{k}

two consecutive terms have at least

2 k

common points.
4. Let us attach to each

x \in \dot{E}

a neighborhood

V_{x}^{k}

and either

Q

all of these neighborhoods. If

a, b \in E

the almost-closure

\dot{E}

coincides with the closure

\overset{―}{E}

of E , so is a closed set. In this case, we can apply the Bore1-Lebesgue theorem and choose in Q a finite number of terms covering entirely the set E, therefore a fortiori the set E. These terms can obviously be arranged in a sequence in such a way that two consecutive terms are not separated by E . Taking into account Lemma II, we deduce the

Lemma III. If

a, b \in E

and if

V

is a set of neighborhoods

V_{*}^{k}

corresponding to all points

x

of

\dot{E} = \overset{―}{E}

, we can choose a finite number of terms in Q),

V_{x_{1}}^{k}, V_{x_{2}}^{k}, \dots, V_{x_{m}}^{k}

completely covering the set E and two consecutive ones

V_{x_{i}}^{k}, V_{x_{i + 1}}^{k}

having at least

2 k

common points.
5. A function

f = f (x)

, uniform and defined on any linear set E is said to be convex, non-concave, polynomial, non-convex or concave of order

n

on E if inequality
(2)

[x_{1}, x_{2}, \dots, x_{n + 2}; f] >, \geq, =, \leq ou < 0

is satisfied whatever

x_{1}, x_{2}, \dots, x_{n + 2} \in E

.
All these functions are order functions

n (^{1})

.

Any convex, non-concave, etc. function of order

n

on E is still convex, non-concave,... etc. of order

n

on any subset of E.

We recall that the necessary and sufficient condition for

f

, defined on a finite set

\begin{matrix} (3) & x_{1} < x_{2} < \dots < x_{m}, m \geq n + 2 \end{matrix}

either convex, non-concave, etc. of order

n

is that we have

\begin{matrix} (4) & [x_{i}, x_{i + 1}, \dots, x_{i + n + 1}; f] >, \geq, =, \leq ou < o \\ i = 1, 2, \dots, m - n - 1 \end{matrix}

Cette propriété résulte du fait que toute différence divisée sur

n + 2

points de (3) est une moyenne arithmétique des différences divisées spécifiées par l'inégalité (4), donc

\begin{matrix} [x_{i_{1}}, x_{i_{2}}, \dots, x_{i_{n + 2}}; t] = \sum_{i = 1}^{n - n - 1} A_{i} [x_{i}, x_{i + 1}, \dots, x_{i + n + 1}; f] \\ A_{i} \geq 0, i = 1, 2, \dots, m - n - 1, \sum_{i = 1}^{m - n - 1} A_{i} = 1 \end{matrix}

Si

i_{1} < i_{2} < \dots < i_{n + 2}

on a d'ailleurs surement

A_{i_{1}} > 0, A_{i_{n + 2} - n - 1} > 0

. Les

A_{i}

sont indépendants de la fonction

/̸

.

De cette propriété nous déduisons, en particulier, que:
Lemme IV. Si une fonction

f

est convexe, non-concave, . . . etc. d'ordre

n

sur deux sections

E_{1}, E_{2}

de

E_{1}

ayant au moins

n + I

points communs elle est convexe, non-concave,... etc. d'ordre n sur la réunion des ensembles

F_{1}, F_{2}

.

Ceci résulte immédiatement de ce qui précède et du fait que si

α_{1}, α_{2}

,

\dots, α_{n + 2}

sont

n + 2

points de la réunion de

E_{1}, E_{2}

et

β_{1}, β_{2}, \dots, β_{n + 1}

,

n +

I points communs à

E_{1}

et

E_{2}

, les points

α_{i}

,

β_{i}

rangés dans 1'ordre croissant jouissent de la propriété que

n + 2

points consécutifs quelconques appartiennent tous à

E_{1}

ou à

F_{2}

.
6. Introduisons maintenant la définition suivante:

Définition I. La fonction

f

est dite localement convexe, non-concave, ... etc. d'ordre

n

sur E si à tout

x \in \dot{E}

correspond un voisinage

V_{x}^{k}

ờ la fonction est convexe, non-concave, ... etc. d'ordre

n

.

Nous supposons toujours

n \geq

I. Pour que la définition précédente ait un sens précis il faut que.

E

ait au moins

n + 2

points et que l'on ait

2 k \geq n + I

. La plus petite valeur de

k

qu'on peut ainsi admettre est donc

[\frac{n + 2}{2}]

, en désignant, comme d'habitude, par

[α]

le plus grand entier compris dans

α

.

Nous avons maintenant la propriété suivante:
Théorème I. Toute fonction localement convexe, non-concave..., etc. d'ordre

n

sur E , avec

k = [\frac{3 + 2}{2}]

, est convexe, non-concave, . . etc. d'ordre

n

sur E .

Il suffit de démontrer la propriété pour une section de E contenant ses extrémités. La propriété résulte alors des lemmes III et IV. Dans le cas d'un intervalle les voisinages peuvent être pris au sens ordinarie et la propriété a été donnée alors pour

n = 1

par M. J. B 1aquier

^{1}

).

On peut facilement voir que la considération de la presque-fermeture E dans la définition I est essentielle. Si dans cette définition onplace l'hypothèse

x \in \dot{E}

par l'hypothèse moins restrictive

x \in E

le théo rème I peut ne pas être vrai pour un ensemble qui n'est fermé. Par exemple la fonetic unest pas presque-

f (x) = {\begin{cases} x, & 0 \leq x < I \\ x - I, & I < x \leq 2 \end{cases}

est bien localement polynomiale de tout ordre

n \geq I

avec la nouvelle définition (pour un

k

quelconque), mais n'est pas d'ordre

n

sur son ensemble de définition.

On pourrait encore chercher si on ne peut pas améliorer la propriété par une définition plus restrictive du voisinage. On peut facilement voir que si

n

est pair il suffit de considérer des voisinage ayant au moins

n + 1

points différents de

x

et ayant tous au moins

\frac{n + 2}{2}

points d'une même côté de

x

et au moins

\frac{n}{2}

points de l'autre côté de

x

.
7. On peut aussi imposer à un voisinage d'autres conditions entrenant la convexité. On peut dire, par exemple, que

f

a localement une droite d'appui si, quel que soit le point

x_{0}

de E , différent d'une extrémité

a, b

, il existe un voisinage

V_{x_{0}}^{r}

et une droite non-vérticale

Δ

passing through the point (

x_{0}, f (x_{0})

) leaving the curve

y = f (x)

not below

Δ

For

x \in V_{x_{0}}^{I}

. We then have 1 e

Theorem II. Any function

f

, defined and continuous on the almost closed set E and having locally a support line, is non-concave of order I on F.

The demonstration follows from the facts that every non-concave function of order ia locally has a support line and that this property is not true for a function which is not non-concave of order 1. In

In fact, in this latter case, we can find three points

x_{1} < x_{2} < x_{3}

of E such that

[x_{1}, x_{2}, x_{3}; f] < o

. the set of points where the function

f (x) - - \frac{x - x_{3}}{x_{1} - x_{3}} f (x_{1}) - \frac{x - x_{1}}{x_{3} - x_{1}} f (x_{3})

reaches its maximum (

> 0

) on the part of E included in the closed interval (

x_{1}, x_{3}

), is closed. The ends of this set are points of

E

, different from

a, b

, where there is no local support line.

We demonstrate Theorem III in the same way
. Any function,

f

, defined and continuous on an almost closed set E which is such that, whatever

x_{0} \in E

, different from a and

b

, there are two points

x^{'}, x^{''}, x^{'} < x_{0} < x^{''}

such that if

V_{ε_{0}}^{I}

C

(x^{'}, x^{''})

we can find two points

x_{1}, x_{2}

of

V_{x_{0}}^{1}, x_{1} < x_{0} < x_{2}

verifying the inequality

[x_{0}, x_{1}, x_{2}; f] ≧ 0

, is non-concave of order I on E.

These properties can be generalized further, but there is no point in doing so here.
8. We could also introduce the following definition:

Definition II. The function

f

is locally of order

n

on E, if at all

x \in E

corresponds to a neighborhood

V_{*}^{k}

where the function is of order

n

.

A locally order function

n

is not generally of order

n

on E, however large it may be

k

. For example, the function

f (x) = (x - I)^{n + I}, 0 \leq x \leq I; = 0, I \leq x \leq 2; = - (x - 2)^{n + I}, 2 \leq x \leq 3

is locally of order

n

(regardless of

k

) and yet is not of order

n

in the closed interval ( 0.3 ).

But, we have
Lemma V. If a function

f

is convex or concave of order

n

on two sections

E_{1}, E_{2}

of E having at least

n + 2

common points, it is convex or concave of order

n

on the meeting of the sets

E_{1}, E_{2}

.

This lemma is a consequence of Lemma IV since

f

cannot be convex on one section and concave on the other.

We immediately deduce
Theorem IV. If at all

x \in \dot{E}

corresponds to a neighborhood

V_{x}^{k}

, with

k = [\frac{n + 3}{2}]

where the function

f

is convex or concave of order

n

, this function is convex or concave of order

n

on E.

Here again the property can be improved by a more restrictive definition of the neighborhood if

n

is odd. It is then sufficient to consider neighborhoods having at least

n + 2

different points of

x

and having all at least

\frac{n + 3}{2}

points on the same side of

x

and at least

\frac{n + 1}{2}

points on the other side of

x

.

40 note on smooth convex functions of higher order (viii)

Before finishing, let us make some remarks on the divided differences of a function $f$ . Let us pose.

\begin{aligned} {\bar{Δ}}_{n} = & max_{(E)} [x_{1}, x_{2}, \dots, x_{n + 1}; f], Δ_{n} = min_{(E)} [x_{1}, x_{2}, \dots, x_{n + 1}; f] \\ Δ_{n} = max_{(E)} | [x_{1}, x_{2}, \dots, x_{n + 1}; f] | = max (| {\bar{Δ}}_{n} |, | Δ_{n} |) \end{aligned}

Finite or infinite numbers

{\bar{Δ}}_{n}, {\underset{―}{Δ}}_{n}

And

Δ_{n}

are the

n

-th upper bound, the

n

-th lower bound and the

n

-th terminal of

f

on E. We will also designate them by

{\bar{Δ}}_{n} [f; E], {\underset{―}{Δ}}_{n} [f; E]

And

Δ_{n} [f; E]^{1}

).

If we take for E the finite set (3) and we set

Δ_{k}^{i} = [x_{i}, x_{i + 1}, \dots, x_{i + k}; f]

We have

\begin{matrix} {\bar{Δ}}_{n} = max (Δ_{n}^{I}, Δ_{n}^{2}, \dots, Δ_{n}^{m - n}), Δ_{n} = min (Δ_{n}^{I}, Δ_{n}^{2}, \dots, Δ_{n}^{m - n}) \\ Δ_{n} = max (| Δ_{n}^{I} |, | Δ_{n}^{2} |, \dots, | Δ_{n}^{m - n} |) . \end{matrix}

Let us also agree to note by

E_{1} + E_{2}

the meeting of two sets

E_{1}

And

E_{2}

, we then have the

Lemma VI. If

E_{1}, E_{2}

are two sections of E having at least

n

common points, we have

\begin{aligned} {\bar{Δ}}_{n} [f; E_{1} + E_{12}] = max ({\bar{Δ}}_{n} [f; E_{1}], {\bar{Δ}}_{n} [f; E_{2}]) \\ {\underset{―}{Δ}}_{n} [f; E_{1} + E_{2}] = min ({\underset{―}{Δ}}_{n} [f; E_{1}], {\underset{―}{Δ}}_{n} [f; E_{2}]) \\ Δ_{n} [f; E_{1} + E_{2}] = max (Δ_{n} [f; E_{1}], Δ_{n} [f; E_{2}]) . \end{aligned}

Let us prove the third equality. If E ; CE , we obviously have

Δ_{n} [t; E^{*}] \leq Δ_{n} [t; E] .

So we have
(5)

Δ_{n} [f; E_{1} + E_{2}] \geq max (Δ_{n} [f; E_{1}], Δ_{n} [f; E_{2}])

.

Let

α_{1}, α_{2}, \dots, α_{n + 1}, n + I

points of

E_{1} + E_{2}

And

β_{1}, β_{2}, \dots, β_{n} n

common points to

E_{1}

And

E_{2}

. If we arrange the points

β_{i}, α_{i}

in a sequence (3),

n + I

consecutive points always belong to

E_{1}

or to

E_{2}

. We deduce that
done

| [α_{1}, α_{2}, \dots, α_{n + x}; f] | ≦ max (Δ_{n} [f; E_{1}], Δ_{n} [f; E_{2}])

(6)

Δ_{n} [t; E_{1} + E_{2}] ≲ max (Δ_{n} [t; E_{1}], Δ_{n} [t; E_{2}]) .

^{1}

) The numbers

{\bar{Δ}}_{n}, {\underset{―}{Δ}}_{n}, Δ_{n}

may be infinite, we use the usual conventions on operations with signs

\pm \infty

. See, e.g., C. Carath éodory, Vorlesüngen über reelle Funktionen, pp. 14, 15.

The two inequalities (5), (6) prove the property. The first two equalities of the lemma are proved in exactly the same way.

We can now state
Theorem V. If f is a function defined on E, (bounded), we can find three points

x_{0}, x_{1}, x_{2}

(distinct or not) from the closure

\overset{―}{E}

of E so that, whatever the neighborhoods

V_{x_{0}}^{k}, V_{x_{1}}^{k}, V_{x_{1}}^{k}

, with

k = [\frac{n + 1}{2}]

, we have

\begin{aligned} {\bar{Δ}}_{n} [f; V_{x_{0}}^{k}] = {\bar{Δ}}_{n} [f; E] \\ {\underset{―}{Δ}}_{n} [f; V_{x_{1}}^{k}] = Δ_{n} [f; E] \\ Δ_{n} [f; V_{x_{2}}^{k}] = Δ_{n} [f; E] \end{aligned}

Let us demonstrate, for example, the last equality. If the equality were not true, we could attach to each

x \in \overset{―}{E}

a neighborhood

V_{x}^{k}

Or

Δ_{n} [f; V_{x}^{k}] << A_{n} [f; E]

. Lemmas III and VI show us that this is impossible. We prove the first two inequalities in the same way. It goes without saying that we always assume

n \geq I

.

We have already reported this property, for

Δ_{n}

assumed to be finite, in the case where E is everywhere dense in

(a, b)^{1}

) and also when

Δ_{n}

is infinite under certain restrictions

^{2}

).

In the theorem

V

The definition of neighborhood can be modified in various ways, but we are not concerned with this question here.

Cernăuti, July 8, 1939.

$^{1}$ ) In note VI we gave a slightly different definition of section. In this note E , was always closed and we only needed closed sections of $E_{.}$ .
$^{1}$ ) This is sufficient for our considerations. In reality the two neighborhoods have at least $2 k +$ I common points and even always an infinity if $x \in E^{'}$ .
$^{1)}$ For the notations and properties of these functions see my previous work.
$^{1}$ ) J. B1aquier, Sobve dos condiciones carateristicas de las functiones convexas, Atti Congresso Bologna, 2, 349-353 (1930).
$^{1}$ ) Tiberiu Popoviciu, On some properties of functions of one or two real variables. Thesis, Paris 1933 or Mathematica, 8, 1-85 (1934), sp. p. 10.
$^{2}$ ) Tiberiu Popoviciu, Notes on Higher Order Convex Functions (I). Mathematica, 12, 81-92 (1936), sp. p. 89.

1939

Notes on Higher Order Convex Functions (VIII)

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ROMANIAN ACADEMY

BULLETIN OF THE SCIENTIFIC SECTION

NOTE ON HIGHER ORDER CONVEX FUNCTIONS (VIII)

BY

T'IBERIU POPOVICIU

40 note on smooth convex functions of higher order (viii)

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